A247904 Start with a single pentagon; at n-th generation add a pentagon at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

1, 6, 21, 56, 131, 286, 601, 1236, 2511, 5066, 10181, 20416, 40891, 81846, 163761, 327596, 655271, 1310626, 2621341, 5242776, 10485651, 20971406, 41942921, 83885956, 167772031, 335544186, 671088501, 1342177136, 2684354411, 5368708966, 10737418081

Offset

0,2

Comments

Refer to A247619, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e."vertex to side" expansion version. Let us assign the label "1" to the pentagon at the origin; at n-th generation add a pentagon at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping pentagons will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The pentagons count is A005891. See illustration. For n >= 1, (a(n) - a(n-1))/5 is A000225.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Kival Ngaokrajang, Illustration of initial terms

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Formula

a(0) = 1, for n >= 1, a(n) = 5*A000225(n) + a(n-1).

a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). - Colin Barker, Sep 26 2014

G.f.: (1+2*x+2*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Sep 26 2014

From G. C. Greubel, Feb 18 2022: (Start)

a(n) = 10*2^n - (5*n + 9).

E.g.f.: 10*exp(2*x) - (9 + 5*x)*exp(x). (End)

Mathematica

LinearRecurrence[{4, -5, 2}, {1, 6, 21}, 51] (* G. C. Greubel, Feb 18 2022 *)

Prog

(PARI)
a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1))
for (n=0, 50, print1(a(n), ", "))
(PARI)
Vec(-(2*x^2+2*x+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014
(Magma) [10*2^n -(5*n+9): n in [0..50]]; // G. C. Greubel, Feb 18 2022
(Sage) [5*2^(n+1) -(5*n+9) for n in (0..50)] # G. C. Greubel, Feb 18 2022

Crossrefs

Cf. Vertex to vertex version: A061777, A247618, A247619, A247620.

Cf. Vertex to side version: A101946, A247903, A247905.

Cf. A000225, A005891.

Sequence in context: A337895 A145134 A256571 * A074745 A296821 A056414

Adjacent sequences: A247901 A247902 A247903 * A247905 A247906 A247907

Keyword

nonn,easy

Author

Kival Ngaokrajang, Sep 26 2014

Extensions

More terms from Colin Barker, Sep 26 2014

Status

approved